Fluid Mechanics: Pascal, Bernoulli & Viscosity

Fluid Statics: Pressure in Fluids

A fluid is any substance that can flow — liquids and gases. Pressure at a point in a fluid at rest is defined as the normal force per unit area:

$P = \frac{F}{A}$

• $P$: pressure (Pa or N m$^{-2}$)
• Dimensional formula: [M L$^{-1}$ T$^{-2}$]
• Pressure is a scalar quantity — it acts equally in all directions at a point (Pascal's law for fluids at rest)

Pressure Variation with Depth

$P = P_0 + \rho g h$

• $P_0$: atmospheric pressure at surface ($1.013 \times 10^5$ Pa = 1 atm)
• $\rho$: density of fluid (kg m$^{-3}$)
• $h$: depth below the surface (m)
• Pressure depends only on depth, not on shape or size of the container

Pascal's Law

A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container.

Hydraulic Press/Lift: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

Mechanical advantage: $\frac{F_2}{F_1} = \frac{A_2}{A_1}$

A small force on a small piston produces a large force on a large piston. Work done is conserved: $F_1 d_1 = F_2 d_2$.

Archimedes' Principle and Buoyancy

Any body partially or completely submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced:

$F_B = \rho_f V_{\text{submerged}} g$

• $\rho_f$: density of fluid
• $V_{\text{submerged}}$: volume of body below the fluid surface

Conditions for floating/sinking:

• Floats: $\rho_{\text{body}} < \rho_{\text{fluid}}$ (partially submerged)
• Neutral: $\rho_{\text{body}} = \rho_{\text{fluid}}$ (fully submerged, equilibrium)
• Sinks: $\rho_{\text{body}} > \rho_{\text{fluid}}$

Fraction submerged (floating body): $\frac{V_{\text{sub}}}{V_{\text{total}}} = \frac{\rho_{\text{body}}}{\rho_{\text{fluid}}}$

Fluid Dynamics: Equation of Continuity

For an ideal (incompressible, non-viscous) fluid in steady flow:

$A_1 v_1 = A_2 v_2$

• $A$: cross-sectional area (m$^2$)
• $v$: flow velocity (m s$^{-1}$)
• Product $Av$ = volume flow rate $Q$ (m$^3$ s$^{-1}$)

Bernoulli's Theorem

For an ideal fluid in steady, streamlined flow:

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$

• $P$: static pressure (Pa)
• $\frac{1}{2}\rho v^2$: dynamic pressure (kinetic energy per unit volume)
• $\rho g h$: hydrostatic pressure (potential energy per unit volume)

This is essentially conservation of energy per unit volume along a streamline.

Applications of Bernoulli's Theorem

Torricelli's theorem (speed of efflux): $v = \sqrt{2gh}$ where $h$ is the depth of the orifice below the free surface. The stream acts as a projectile after exit.

Venturi meter: Measures flow speed using pressure difference at a constriction: $v_1 = A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho(A_1^2 - A_2^2)}}$

Lift on an aerofoil: Faster air over the top surface (lower pressure) and slower air below (higher pressure) create net upward force.

Viscosity

Viscosity is the internal friction in a fluid that opposes relative motion between adjacent layers.

Newton's law of viscosity: $F = -\eta A \frac{dv}{dy}$

• $\eta$: coefficient of dynamic viscosity (Pa s or N s m$^{-2}$)
• $dv/dy$: velocity gradient (s$^{-1}$)
• Dimensional formula of $\eta$: [M L$^{-1}$ T$^{-1}$]
• 1 Poise = 0.1 Pa s (CGS unit)

Stokes' Law

A sphere of radius $r$ moving through a viscous fluid at velocity $v$ experiences a drag force:

$F_{\text{drag}} = 6\pi \eta r v$

Terminal Velocity

When drag force + buoyancy = weight:

$v_T = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}$

• $v_T$: terminal velocity (m s$^{-1}$)
• $\rho_s$: density of sphere
• $\rho_f$: density of fluid

Reynolds Number

$Re = \frac{\rho v D}{\eta}$

• $D$: characteristic dimension (pipe diameter)
• $Re < 2000$: laminar flow
• $Re > 4000$: turbulent flow
• $2000 < Re < 4000$: transition

Poiseuille's Equation (Flow through a pipe)

$Q = \frac{\pi \Delta P r^4}{8 \eta L}$

• $Q$: volume flow rate
• $\Delta P$: pressure difference
• $r$: pipe radius
• $L$: pipe length

---